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Theorem isperf3 20862
Description: A perfect space is a topology which has no open singletons. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = 𝐽
Assertion
Ref Expression
isperf3 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem isperf3
StepHypRef Expression
1 lpfval.1 . . 3 𝑋 = 𝐽
21isperf2 20861 . 2 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
3 dfss3 3578 . . . 4 (𝑋 ⊆ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝑋))
41maxlp 20856 . . . . . 6 (𝐽 ∈ Top → (𝑥 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑥𝑋 ∧ ¬ {𝑥} ∈ 𝐽)))
54baibd 947 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑋) ↔ ¬ {𝑥} ∈ 𝐽))
65ralbidva 2984 . . . 4 (𝐽 ∈ Top → (∀𝑥𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
73, 6syl5bb 272 . . 3 (𝐽 ∈ Top → (𝑋 ⊆ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
87pm5.32i 668 . 2 ((𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)) ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
92, 8bitri 264 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wss 3560  {csn 4153   cuni 4407  cfv 5850  Topctop 20612  limPtclp 20843  Perfcperf 20844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-top 20616  df-cld 20728  df-ntr 20729  df-cls 20730  df-lp 20845  df-perf 20846
This theorem is referenced by:  perfi  20864  perfopn  20894  t1connperf  21144
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